Global Carleman estimates for degenerate parabolic operators with applications:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
January 2016
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| Series: | Memoirs of the American Mathematical Society
volume 239, number 1133 (5th of 6 numbers) |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | ix, 209 Seiten |
| ISBN: | 9781470414962 |
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MARC
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| 245 | 1 | 0 | |a Global Carleman estimates for degenerate parabolic operators with applications |c P. Cannarsa, P. Martinez, J. Vanconstenoble |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c January 2016 | |
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Record in the Search Index
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| adam_text | Contents
Chapter 1. Introduction 1
1.1. Stochastic invariance 2
1.2. Laminar flow 3
1.3. Budyko-Sellers climate models 3
1.4. Fleming֊’Viot gene frequency model 4
1.5. Null controllability in one space dimension 4
Part 1. Weakly degenerate operators with Dirichlet boundary
conditions 9
Chapter 2. Controllability and inverse source problems: Notation and main
results 11
2.1. Notation and assumptions 11
2.1.1. Geometric assumptions and properties of domains 11
2.1.2. Assumptions on degeneracy 12
2.2. Statement of the controllability problem and main results 14
2.3. Statement of the inverse source problems and main results 15
2.3.1. First inverse source problem 16
2.3.2. Second inverse source problem 17
2.4. Estimates with respect to the degeneracy parameter a 18
Chapter 3. Global Carlem n estimates for weakly degenerate operators 19
3.1. Function spaces and well-posedness 19
3-1.1. Function spaces 19
3.1.2. Trace theory and integration-by-parts formula 20
3.1.3. Regularity results 21
3.1.4. WelLposedness 22
3.2. Observability: inequality and cost 22
3.3. Fundamental tools 23
3-3.1- Improved Hardy type inequalities 23
3.3.2. Topological lemmas 24
3-4. Global Carlemaa estimates for weakly degenerate operators 25
3.4.1. Statement of the global Carleman estimate 25
3-4.2. Comparison with the literature on Carleman estimates 27
3.4.3. Additional remarks 28
3.5. Extensions 28
3.5.1. Degenerate parabolic operators with lower order terms 28
3.5.2. Weakened geometric assumptions 29
Chapter 4. Some Hardy-type inequalities (proof of Lemma 3.18} 31
Ш
IV
CONTENTS
4.1. Hardy-type inequalities in space dimension 1 31
4.1.1. The well known Haxdy-type inequality in space dimension 1 31
4.1.2. An improved version: proof of Lemma 3.16 31
4.2. Hardy-type inequalities in space dimension 2 34
4.2.1. Extension of the classical Hardy inequality under Hyp. 2.4. 34
4.2.2. Consequence: proof of Lemma 3.18 under Hyp. 2.4. 37
4.2.3. Consequence: proof of Lemma 3.18 under Hyp. 2.2. 37
Chapter 5. Asymptotic properties of elements of n 45
5.1. Asymptotic behavior near the boundary of the elements of
and n H t0{Q) under Hyp. 2.4 45
5.1.1. Statement of the main asymptotic properties 45
5.1.2. Proof of Lemma 5.2 46
5.1.3. Proof of Lemma 5.1 48
5.2. Asymptotic properties under Hyp֊ 2.2 62
Chapter 6. Proof of the topological lemma 3.21 63
6.1. Preliminary Lemma 63
6.2. Proof of Lemma 6.1 63
6.3. Proof of Lemma 3.21 66
Chapter 7. Outlines of the proof of Theorems 3.23 and 3.26 67
7.1. Outlines of the proof of Theorems 3.23 and 3.26 under Hyp. 2.4 57
7.1.1. Under Hyp. 2.4: choice of the weight functions and objectives 57
7.1.2. Step 1 (under Hyp. 2.4): computation of the scalar product on
subdomains fl*5 69
7.1.3. Step 2 (under Hyp. 2.4): an estimate of the scalar product on
subdomains Cl6 61
7.1.4. Step 3 (under Hyp. 2.4): the limits as Cls — £2 61
7.1.5. Step 4 (under Hyp. 2.4): partial Garleman estimate 62
7.1.6. Step 5 (under Hyp. 2.4): from the partial to the global Car lem an
estimate 62
7.1.7. Step 6 (under Hyp. 2.4): global Oarleman estimates 64
7,2. Generalization: main changes under Hyp, 2.2 65
7.2.1. The choice of the weight functions under Hyp. 2.2 65
7.2.2. Step 1 (under Hyp. 2.2): computation of the scalar product on
subdomains 66
7.2.3. Step 2 (under Hyp. 2.2): estimates for the distributed terms 66
7-2,4. Step 3 (under Hyp. 2.2): the limits iV —* fi 67
7.2.5. Step 4 (under Hyp. 2.2): partial Oarleman estimate 67
7.2.6. Steps 5 and 6 (under Hyp. 2.2): from the partial to the global
Carleman estimate 67
Chapter 8. Step 1: computation of the scalar product on subdomains (proof
of Lemmas 7.1 and 7.16) 69
8.1. The scalar product under Hyp. 2.4 69
8.2. The scalar product under Hyp. 2,2 73
Chapter 9. Step 2: a first estimate of the scalar product; proof of Lemmas
7.2, 7.4, 7.18 and 7.19
75
CONTENTS
V
9.1. A first estimate of the scalar product under Hyp. 2.4: proof of
Lemmas 7.2 and 7.4 75
9.1.1. Estimate of the first order terms DTX: proof of Lemma 7.2 75
9.1.2. Estimate of the zero order term DTģ: proof of Lemma 7.4 80
9.2. A first estimate of the scalar product under Hyp. 2.2: proof of
Lemmas 7.18 and 7.19 83
9.2.1. A general result about the asymptotic behaviour near the boundary:
proof of Lemma 7.17 83
9.2.2. First consequence: estimate of the first order term DTX (proof of
Lemma 7.18) 84
9.2.3. Second consequence: estimate of the zero order term DT0 (proof of
Lemma 7.19) 86
Chapter 10. Step 3: the limits as fl5 — (proof of Lemmas 7.5 and 7.20) 89
10.1. The limits as 5 —» ft under Hyp. 2.4 (proof of Lemma 7.5) 89
10.1.1. Statement of the convergence results 89
10.1.2. Convergence of the distributed terms: proof of Lemma 10.1 89
10.1.3. Convergence of the boundary term: proof of Lemma 10.2 93
10.1.4. An identity of interpolation type 96
10.2. The limits as -u fl under Hyp. 2.2 (proof of Lemma 7.20) 98
10.2.1. Statement of the convergence results 98
10.2.2. Ideas of the proof of Lemmas 10.7 and 10.8 99
Chapter 11. Step 4: partial Carleman estimate (proof of Lemmas 7.6 and
7.21) 101
11.1. The partial Carleman estimate under Hyp. 2.4 (proof of Lemma 7.6) 101
11.1.1. The consequence of the estimate of the scalar product given in
Lemma 7.5 101
11.1.2. Some adapted Hardy-type inequalities 102
11.1.3. Gonsequence: proof of Lemma 7.6 105
11.2. The partial Carleman estimate under Hyp. 2.2 (proof of Lemma
7.21) 112
11.2.1. The consequence of the estimate of the scalar product given in
Lemma 7.20 112
11.2.2. Proof of Lemma 7,21 113
11.2.3. Uniform estimates when ot —» 1՜ (under Hyp. 2.12) 115
Chapter 12. Step 5: from the partial to the global Carleman estimate (proof
of Lemmas 7.9-7.11) 117
12.1. Estimate of the zero order term: proof of Lemma 7.8 117
12.2. Estimate of the first order spatial derivatives: proof of Lemma 7.9 117
12.2.1. The non uniform estimate 117
12.2.2. The uniform estimate 119
12.3. Estimate of the second order spatial derivatives: proof of Lemma
7.10 120
12.4. Estimate of the first order time derivative: proof of Lemma 7-11 121
Chapter 13. Step 6: global Carleman estimate (proof of Lemmas 7.12, 7.14
and 7.15)
123
VI
CONTENTS
13.1. The global Carleman estimate for z: proof of Lemma 7.12 123
13.2. The first global Carleman estimate for w: proof of Lemma 7.14 125
13.2.1. Zero order term estimates. 125
13.2.2. First order spatial derivatives estimates. 125
13.2.3. First order time derivative estimate. 127
13.2.4. Second order spatial derivatives estimates. 127
13.2.5. Conclusion: proof of Lemma 7.14. 128
13.3. The second global Carleman estimate for w: proof of Lemma 7.15 128
Chapter 14. Proof of observability and controllability results 131
14.1. Proof of Theorem 3.13 131
14.2. Equivalence between null controllability and observability 132
Chapter 15. Application to some inverse source problems: proof of Theorems
2.9 and 2.11 135
15.1. Proof of Theorem 2.9 135
15.2. Proof of Theorem 2.11 139
Part 2. Strongly degenerate operators with Neumann boundary
conditions 141
Chapter 16. Controllability and inverse source problems: notation and main
results 143
16.1. Notation and assumptions 143
16.1.1. Geometric assumptions and properties of the domain 143
16.1.2. Assumptions on degeneracy 143
16.2. Statement of the controllability problem and mam results 143
16.2.1. A null controllability result for aG [1,2). 144
16.2.2. Counterexample for a e [2,+oo). 145
16.2.3. Explosion of the controllability cost as a — 2՜ in space dimension
1 146
16.3. Statement of the inverse source problems and main results 147
16.3.1. First inverse source problem 147
16-3,2. Second inverse source problem 148
Chapter 17, Global Carleman estimates for strongly degenerate operators 149
17.1. Functional spaces and welLposedness 149
17.1.1. Function spaces 149
17.1.2. Normal trace theory and integration-by-֊parts formula 149
17.1.3. Regularity results 150
17.1.4. WelLposedness 150
17.2. Observability: inequality and cost 150
17.3. Global Carleman estimates for strongly degenerate operators 151
17.4. Fundamental tools 151
17.5. Some extensions 152
17.5.1. Global Carleman estimate for a more general degenerate parabolic
equation 152
17.5.2. Weakened geometric assumptions 153
Chapter 18. Hardy֊type inequalities: proof of Lemma 17.10 and applications 155
CONTENTS vii
18.1. Some Hardy-type inequalities in dimension 1 155
18.1.1. The classical Hardy inequality when a C (1,2) 155
18.1.2. A first extension of the classical Hardy inequality 156
18.2. Proof of Lemma 17.10 156
18.2.1. Proof of Lemma 17.10, part (i), under Hypothesis 17.9. 157
18.2.2. Proof of Lemma 17.10, part (ii), under Hypothesis 17.9. 157
18.2.3. Proof of Lemma 17.10 under Hypothesis 16.1. 157
18.3. Some Hardy-type inequalities adapted to our problem 157
18.3.1. The natural extension of Lemma 17.10. 157
18.3.2. Consequence of Lemma 18.3: another Hardy-type inequalities. 159
18.3.3. Consequences of Lemma 18.4. 160
Chapter 19. Global Carleman estimates in the strongly degenerate case:
proof of Theorem 17.7 163
19.1. Outlines of the proof of Theorem 17.7 163
19.2. Proof of Theorem 17.7 under Hyp. 17.9 163
19.2.1. Steps 1 and 2 (under Hyp. 17.9): computation and estimate of the
scalar product on subdomains 163
19.2.2֊ Step 3 (under Hyp. 17.9): the limits as Qó — Q 164
19.2.3. Step 4 (under Hyp. 17.9): partial Carleman estimate 167
19.2.4. Steps 5 and 6 (under Hyp. 17.9): from the partial to the global
Carleman estimate (proof of Theorem 17.7) 170
19.3. Proof of Theorem 17.12 under Hyp. 16.1 171
Chapter 20. Proof of Theorem 17.6 (observability inequality) 173
Chapter 21. Lack of null controllability when a 2: proof of Proposition
16.5 177
21.1. The geometrical situation 177
21.2. Proof of Proposition 16.5 177
21.2.1. The problem in polar coordinates 177
21.2.2. The means of v satisfies a one֊dimensional degenerate parabolic
problem 178
21.2.3. Transformation into a nondegenerate parabolic problem set in the
half֊ line 178
21.2.4֊ The reason for which null controllability fails 180
Chapter 22. Explosion of the controllability cost as a —» 2՜ in space
dimension 1: proof of Proposition 16.7 183
22.1. The method to prove the explosion of the controllability cost as
a — ֊ 2՜ in space dimension 1 183
22.2. Bessel functions and their application to our problem 183
22.2.1. Useful properties of Bessel functions 183
22.2.2. The miscelleanous function wa 184
22.2.3. Proof of Proposition 22.1: the properties of the function wa 184
22.3. The underlying eigenvalue problem and the link to Bessel functions 187
22.3.1. The underlying eigenvalue problem 187
22.3.2֊ An estimate for the first eigenvalue 187
22.3.3֊ The expression of the eigenfunction in terms of classical Bessel
functions
188
viii
CONTENTS
Part 3. Some open problems 193
Chapter 23. Some open problems 195
23.1. Boundary control 195
23.2. Other classes of degenerate operators 196
23.3. The Fleming-Viot gene frequency model 196
23.4. Observability and controllability cost 197
23.5. More on inverse problems 197
Bibliography 199
Index 209
|
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| id | DE-604.BV043382902 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:24:25Z |
| institution | BVB |
| isbn | 9781470414962 |
| language | English |
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| physical | ix, 209 Seiten |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spelling | Cannarsa, Piermarco 1957- (DE-588)17369361X aut Global Carleman estimates for degenerate parabolic operators with applications P. Cannarsa, P. Martinez, J. Vanconstenoble Providence, Rhode Island American Mathematical Society January 2016 ix, 209 Seiten txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society volume 239, number 1133 (5th of 6 numbers) Carleman-Methode (DE-588)4288492-5 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 s Carleman-Methode (DE-588)4288492-5 s DE-604 Martinez, Patrick 1970- (DE-588)1082391042 aut Vancostenoble, Judith 1972- (DE-588)1082391239 aut Memoirs of the American Mathematical Society volume 239, number 1133 (5th of 6 numbers) (DE-604)BV008000141 1133 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028801530&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cannarsa, Piermarco 1957- Martinez, Patrick 1970- Vancostenoble, Judith 1972- Global Carleman estimates for degenerate parabolic operators with applications Memoirs of the American Mathematical Society Carleman-Methode (DE-588)4288492-5 gnd Elliptischer Differentialoperator (DE-588)4140057-4 gnd |
| subject_GND | (DE-588)4288492-5 (DE-588)4140057-4 |
| title | Global Carleman estimates for degenerate parabolic operators with applications |
| title_auth | Global Carleman estimates for degenerate parabolic operators with applications |
| title_exact_search | Global Carleman estimates for degenerate parabolic operators with applications |
| title_full | Global Carleman estimates for degenerate parabolic operators with applications P. Cannarsa, P. Martinez, J. Vanconstenoble |
| title_fullStr | Global Carleman estimates for degenerate parabolic operators with applications P. Cannarsa, P. Martinez, J. Vanconstenoble |
| title_full_unstemmed | Global Carleman estimates for degenerate parabolic operators with applications P. Cannarsa, P. Martinez, J. Vanconstenoble |
| title_short | Global Carleman estimates for degenerate parabolic operators with applications |
| title_sort | global carleman estimates for degenerate parabolic operators with applications |
| topic | Carleman-Methode (DE-588)4288492-5 gnd Elliptischer Differentialoperator (DE-588)4140057-4 gnd |
| topic_facet | Carleman-Methode Elliptischer Differentialoperator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028801530&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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